The value of $c$ for the function $f(x) = \log x$ on $[1, e]$ if Lagrange's Mean Value Theorem $(LMVT)$ is applied,is

Let $f:[1,3] \rightarrow R$ be continuous and differentiable in $(1,3)$ such that $f^{\prime}(x)=[f(x)]^2+4$ for all $x \in (1,3)$. Then:

Verify the Mean Value Theorem for the function $f(x) = x^{2}$ in the interval $[2, 4]$.

Let $g: R \rightarrow R$ be a non-constant twice differentiable function such that $g^{\prime}\left(\frac{1}{2}\right)=g^{\prime}\left(\frac{3}{2}\right)$. If a real-valued function $f$ is defined as $f(x)=\frac{1}{2}[g(x)+g(2-x)]$,then:

Let $f(x)$ satisfy all the conditions of the Mean Value Theorem in $[0, 2]$. If $f(0) = 0$ and $|f'(x)| \le \frac{1}{2}$ for all $x$ in $[0, 2]$,then:

Rolle's theorem is applicable for the function $f(x) = x^2 - 4$ in which of the following intervals?